rm(list = ls())
source("C:/Users/mbofi/Dropbox/CeMSIIS/GitHub/Allocation/case-study/aux_functions.R") #local
# setwd("C:/Users/mbofi/Dropbox/CeMSIIS/GitHub/Allocation/case-study")#local
# setwd("~/GitHub/Allocation/case-study")

library(tidyverse)
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library(dplyr)
library(plyr)
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#> ------------------------------------------------------------------------------
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library(NCC)
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#>   method               from 
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library(mmtable2)
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#>     table1
library(gt)
#> Warning: package 'gt' was built under R version 4.2.1
# OPTIMAL ALLOCATION USING OLD NOTATION
##########################################
# Simulation function
# modification of sim_designs() function
# also returning the matrix (r_is)
##########################################

sim_designs_rr <- function(r1,r2,mu0,mu1,mu2,N,alloc="sqrt",trend="stepwise",sl=0.2){
  
  r3 = 1-r1-r2
  
  if(r1 == 1){
    
    if(alloc == "one"){
      r22 <- r11 <- r01 <- r1/3
      r02 <- r12 <- r23 <- r03 <- 0
    }
    if(alloc != "one"){ 
      r01 <- sqrt(2)/(2+sqrt(2))
      r22 <- r11 <- (1-r01)/2 
      r02 <- r12 <- r23 <- r03 <- 0
    }
    
  }else{
    
    if(alloc == "opt"){ 
      
      r11 <- r01 <- r1/2
      
      r2_opt <- f(r1=r1,r2=r2)
      r02 <- r2_opt[3]
      r12 <- r2_opt[4]
      r22 <- r2_opt[5]
      
      r23 <- r03 <- r3/2
      
    }
    if(alloc == "one"){
      r11 <- r01 <- r1/2
      r22 <- r12 <- r02 <- r2/3
      r23 <- r03 <- r3/2
    }
    if(alloc == "sqrt"){
      r11 <- r01 <- r1/2
      r02 <- r2*sqrt(2)/(2+sqrt(2))
      r22 <- r12 <- (r2-r02)/2
      r23 <- r03 <- r3/2
    }
    
  }
  # c(r11,r01)
  
  n11 = round(r11*N)
  n01 = round(r01*N)
  
  n22 = round(r22*N)
  n12 = round(r12*N)
  n02 = round(r02*N)
  
  n23 = round(r23*N)
  n03 = round(r03*N)
  
  # c(r11,r01,r22,r12,r02,r23,r03)
  # c(n11,n01,n22,n12,n02,n23,n03)
  
  means = c(mu0,mu1,mu2)
  
  treatment = c(
    sample(c(rep(1,n11),rep(0,n01))),
    sample(c(rep(2,n22),rep(1,n12),rep(0,n02))),
    sample(c(rep(2,n23),rep(0,n03)))
  )
  
  Nsim = length(treatment) 
  
  if(r1==1){
    treatment = sample(treatment)
    period = rep(1,Nsim)
  }else{
    period = c(
      rep(1,n11+n01),
      rep(2,n22+n12+n02),
      rep(3,n23+n03)
    )
  }
  
  if(trend=="stepwise"){
    
    response = rnorm(Nsim,
                     mean=means[treatment[1:Nsim]+1]+sw_trend(cj=period[1:Nsim], lambda=sl),
                     sd=1) 
    
  }
  if(trend=="linear"){
    
    response = rnorm(Nsim,
                     mean=means[treatment[1:Nsim]+1]+linear_trend(j=1:Nsim, lambda=sl, sample_size=c(0,Nsim)),
                     sd=1) 
  }
  
  data = data.frame(response,treatment,period)
  if(r1==1){
    ss = matrix(c(n22,n11,n01,0,n12,n02,n23,0,n03), nrow=3)
  }else{
    ss = matrix(c(0,n11,n01,n22,n12,n02,n23,0,n03), nrow=3)
  }
  r_matrix = matrix(c(0,r11,r01,r22,r12,r02,r23,0,r03), nrow=3)
  r_periods = c(r1,r2,r3)
  return(list(data=data,ss=ss,r_matrix=r_matrix,r_periods=r_periods))
}
# OPTIMAL ALLOCATION USING THE NEW NOTATION
##########################################
eq_p22 <- function(p22,r1=0.1,r2=0.8){(Power(-1 + p22,3)*(-1 + 2*r1))/((-1 + 2*p22)*(-2 + p22*(7 + p22*(-15 + p22*(19 + 2*p22*(-7 + 2*p22))))))-r2}

f_p=Vectorize(function(r1,r2) { 
  p22=uniroot(eq_p22,c(0,r2),r1=r1,r2=r2)$root
  r22 = r2*p22
  # p02 <- 1/(2-2*p22)
  p02 <- 1/(2-2*p22)-p22
  r02 <- p02*r2
  
  p12 <- 1-p02-p22
  r12 <- p12*r2
  # r12 <- r2- r02- r22
  
  sol=c(r1,r2,r22,r12,r02)
  sol
})

1 Solutions case study - Concurrent controls only

1.1 Design 3: three-period design (symmetric design)

##########################################
# Case study
##########################################

# original study - obtained means
mean_control = 17.3/3.5
mean_arm1 = 66.2/3.5
mean_arm2 = 72.3/3.5 

##########################################
# design 3: three-period design (symmetric design)
##########################################
N = 92
N1 = round(N/3)
N3 = round(N/3)
N2 = N-N1-N3
c(N1,N2,N-N1-N2)
#> [1] 31 30 31
alloc_str <- c("one","opt","sqrt")

Optimal allocation using the old parametrization

m <- sim_designs_rr(r1=N1/N,r2=N2/N,
                   mu0=mean_control,mu1=6,mu2=6,
                   N=N,alloc=alloc_str[2],sl=0)
  
  rownames(m$r_matrix)  <- c("Arm 2", "Arm 1", "Control")
  knitr::kable(m$r_matrix, format = "html", caption = paste(alloc_str[2]), 
               col.names = c("Period 1", "Period 2", "Period 3"), 
               row.names=T, 
               digits=3)
opt
Period 1 Period 2 Period 3
Arm 2 0.000 0.096 0.168
Arm 1 0.168 0.096 0.000
Control 0.168 0.135 0.168 
  
  knitr::kable(t(m$r_periods),col.names = c("Period 1", "Period 2", "Period 3"))
Period 1 Period 2 Period 3
0.3369565 0.326087 0.3369565

Optimal allocation using the new parametrization

r_solutions <- f_p(r1=m$r_periods[1], r2=m$r_periods[2])
# rownames(r_solutions[3:5])  <- c("Arm 2", "Arm 1", "Control")
knitr::kable(r_solutions[3:5],col.names = c("Period 2"),row.names=T,digits=3)
Period 2
1 0.096
2 0.096
3 0.135 
knitr::kable(t(c(r_solutions[1:2], 1-sum(r_solutions[1:2]))),col.names = c("Period 1", "Period 2", "Period 3"),digits=3)
Period 1 Period 2 Period 3
0.337 0.326 0.337

1.2 Design 3: three-period design (non-symmetric design)

##########################################
# design 3: three-period design (non-symmetric design)
##########################################
N = 92
N1 = round(N/3)
N2 = round(2*(N-N1)/3)
c(N1,N2,N-N1-N2)
#> [1] 31 41 20

Optimal allocation using the old parametrization

m <- sim_designs_rr(r1=N1/N,r2=N2/N,
                   mu0=mean_control,mu1=6,mu2=6,
                   N=N,alloc=alloc_str[2],sl=0)
  
  rownames(m$r_matrix)  <- c("Arm 2", "Arm 1", "Control")
  knitr::kable(m$r_matrix, format = "html", caption = paste(alloc_str[2]), 
               col.names = c("Period 1", "Period 2", "Period 3"), 
               row.names=T, 
               digits=3)
opt
Period 1 Period 2 Period 3
Arm 2 0.000 0.169 0.109
Arm 1 0.168 0.087 0.000
Control 0.168 0.190 0.109 
  
  knitr::kable(t(m$r_periods),col.names = c("Period 1", "Period 2", "Period 3"),digits=3)
Period 1 Period 2 Period 3
0.337 0.446 0.217

Optimal allocation using the new parametrization

r_solutions <- f_p(r1=m$r_periods[1], r2=m$r_periods[2])
# rownames(r_solutions[3:5])  <- c("Arm 2", "Arm 1", "Control")
knitr::kable(r_solutions[3:5],col.names = c("Period 2"),row.names=T,digits=3)
Period 2
1 0.169
2 0.087
3 0.190 
knitr::kable(t(c(r_solutions[1:2], 1-sum(r_solutions[1:2]))),col.names = c("Period 1", "Period 2", "Period 3"),digits=3)
Period 1 Period 2 Period 3
0.337 0.446 0.217

2 Numerical examples - Concurrent and non-concurrent controls

Optimal allocation using the old parametrization

# code used in the shiny app
source("C:/Users/mbofi/Dropbox/CeMSIIS/GitHub/Allocation/optimisation/comparisons/aux_functions_ncc.R")
#> Warning: package 'nloptr' was built under R version 4.2.3
#> Warning: package 'latex2exp' was built under R version 4.2.1
#>      [,1] [,2]     [,3]      [,4]
#> [1,]  0.2  0.5 0.376248 0.2096949

results <-optr1(r1=0.3)
colnames(results)=c("r1","r2","r12","r22")
results=as.data.frame(results)
results$r02 <- 1- results$r12-results$r22
results[c(51),]
#>     r1  r2       r12       r22       r02
#> 51 0.3 0.7 0.1656498 0.4318298 0.4025203
results[c(51),3:5]*0.7
#>          r12       r22       r02
#> 51 0.1159549 0.3022809 0.2817642

Optimal allocation using the old parametrization

sol_r22 <- function(r1,r2){
  a <- Power(8 + 36*r1 - 108*Power(r1,2) + 27*Power(r1,3) +
    6*sqrt(3)*sqrt(r1*(16 - 72*r1 + 108*Power(r1,2) - 27*Power(r1,3))),
   0.3333333333333333)

  b <- Power(-2 + a,2) + 6*(-4 + a)*r1 + 9*Power(r1,2)

  sol <- -r1 + (sqrt((b*(-1 + r1))/a) - sqrt(-(((-1 + r1)*
            (-12*sqrt(3)*Power(a,1.5)*sqrt(b*(-1 + r1))*r1 +
              b*(4 + 8*a + Power(a,2) - 12*(2 + a)*r1 + 9*Power(r1,2))))/(a*b))))/
    (4.*sqrt(3))

  sol_p <- (12*r1 + sqrt(3)*(-sqrt((b*(-1 + r1))/a) +
        sqrt(-(((-1 + r1)*(-12*sqrt(3)*Power(a,1.5)*sqrt(b*(-1 + r1))*r1 +
                b*(4 + 8*a + Power(a,2) - 12*(2 + a)*r1 + 9*Power(r1,2))))/(a*b)))))/
   (12.*(-1 + r1))

  return(c(a,b,sol,sol_p))
  
  # 1 - r1 + sqrt(((-1 + r1)*(9*Power(r1,2) + 6*r1*(-4 + Power(8 + 36*r1 - 108*Power(r1,2) + 27*Power(r1,3) + 6*sqrt(3)*sqrt(r1*(16 - 72*r1 + 108*Power(r1,2) - 27*Power(r1,3))),
  #             0.3333333333333333)) + Power(-2 + Power(8 + 36*r1 - 108*Power(r1,2) + 27*Power(r1,3) + 6*sqrt(3)*sqrt(r1*(16 - 72*r1 + 108*Power(r1,2) - 27*Power(r1,3))),
  #            0.3333333333333333),2)))/Power(8 + 36*r1 - 108*Power(r1,2) + 27*Power(r1,3) + 6*sqrt(3)*sqrt(r1*(16 - 72*r1 + 108*Power(r1,2) - 27*Power(r1,3))),0.3333333333333333))/
  #   (4.*sqrt(3)) - sqrt(-7.333333333333333 + 8*Power(-1 + r1,2) + (43*r1)/3. - 7*Power(r1,2) - 
  #     ((-1 + r1)*(4 - 24*r1 + 9*Power(r1,2)))/
  #      (12.*Power(8 + 36*r1 - 108*Power(r1,2) + 27*Power(r1,3) + 6*sqrt(3)*sqrt(r1*(16 - 72*r1 + 108*Power(r1,2) - 27*Power(r1,3))),0.3333333333333333)) - 
  #     ((-1 + r1)*Power(8 + 36*r1 - 108*Power(r1,2) + 27*Power(r1,3) + 6*sqrt(3)*sqrt(r1*(16 - 72*r1 + 108*Power(r1,2) - 27*Power(r1,3))),0.3333333333333333))/12. + 
  #     (sqrt(3)*Power(-1 + r1,2)*r1*Power(8 + 36*r1 - 108*Power(r1,2) + 27*Power(r1,3) + 6*sqrt(3)*sqrt(r1*(16 - 72*r1 + 108*Power(r1,2) - 27*Power(r1,3))),0.16666666666666666))/
  #      sqrt((-1 + r1)*(9*Power(r1,2) + 6*r1*(-4 + Power(8 + 36*r1 - 108*Power(r1,2) + 27*Power(r1,3) + 6*sqrt(3)*sqrt(r1*(16 - 72*r1 + 108*Power(r1,2) - 27*Power(r1,3))),
  #              0.3333333333333333)) + Power(-2 + Power(8 + 36*r1 - 108*Power(r1,2) + 27*Power(r1,3) + 6*sqrt(3)*sqrt(r1*(16 - 72*r1 + 108*Power(r1,2) - 27*Power(r1,3))),
  #             0.3333333333333333),2))))/2.
  
  
}

sol_r12 <- function(r1,r22){
  (1 - r1 - sqrt(1 - r1 - 4*r22 + 4*r1*r22 + 4*Power(r22,2)))/2.
}

sol_r22(r1=.3,r2=.7)
#> [1]  2.7275108 -0.9512085 -0.6977191 -0.9967416
# sol_r12(r1=.3,r22=sol_r22(r1=.3,r2=.7)[1])
sol_r12(r1=.3,r22=sol_r22(r1=.3,r2=.7))
#> [1] -2.0385263 -0.9712281 -0.7224809 -1.0160940
 

Center for Medical Statistics, Informatics and Intelligent Systems, Medical University of Vienna.

[Klassifizierung: intern]

Marta Bofill Roig

marta.bofillroig@meduniwien.ac.at